1 00:00:00,000 --> 00:00:00,310 2 00:00:00,310 --> 00:00:02,990 In this video we're going to do a couple of examples that 3 00:00:02,990 --> 00:00:06,179 deal with parallel and perpendicular lines. 4 00:00:06,179 --> 00:00:12,960 So you have parallel, you have perpendicular, and, of course, 5 00:00:12,960 --> 00:00:16,129 you have lines that are neither parallel nor 6 00:00:16,129 --> 00:00:16,950 perpendicular. 7 00:00:16,949 --> 00:00:19,119 And just as a bit of review, if you've never seen this 8 00:00:19,120 --> 00:00:22,510 before, parallel lines, they never intersect. 9 00:00:22,510 --> 00:00:25,170 So let me draw some axes. 10 00:00:25,170 --> 00:00:27,810 So if those are my coordinate axes right there, that's my 11 00:00:27,809 --> 00:00:31,359 x-axis, that is my y-axis. 12 00:00:31,359 --> 00:00:35,070 If this is a line that I'm drawing in magenta, a parallel 13 00:00:35,070 --> 00:00:38,480 line might look something like this. 14 00:00:38,479 --> 00:00:40,709 It's not the exact same line, but they have 15 00:00:40,710 --> 00:00:42,530 the exact same slope. 16 00:00:42,530 --> 00:00:46,149 If this moves a certain amount, if this change in y 17 00:00:46,149 --> 00:00:49,070 over change in x is a certain amount, this change in y over 18 00:00:49,070 --> 00:00:51,000 change in x is the same amount. 19 00:00:51,000 --> 00:00:53,619 And that's why they never intersect. 20 00:00:53,619 --> 00:00:55,029 So they have the same slope. 21 00:00:55,030 --> 00:00:58,670 22 00:00:58,670 --> 00:01:01,289 Parallel lines have the same slope. 23 00:01:01,289 --> 00:01:03,990 Perpendicular lines, depending on how you want to view it, 24 00:01:03,990 --> 00:01:06,640 they're kind of the opposite. 25 00:01:06,640 --> 00:01:08,609 Let's say that this is some line. 26 00:01:08,609 --> 00:01:11,299 A line that is perpendicular to that will not only 27 00:01:11,299 --> 00:01:17,340 intersect the line, it will intersect it at a right angle, 28 00:01:17,340 --> 00:01:20,859 at a 90 degree angle. 29 00:01:20,859 --> 00:01:23,420 And I'm not going to prove it for you here. 30 00:01:23,420 --> 00:01:26,269 I actually prove it in the linear algebra playlist. But a 31 00:01:26,269 --> 00:01:29,409 perpendicular line's slope-- so let's say that this one 32 00:01:29,409 --> 00:01:34,109 right here, let's say that yellow line has a slope of m. 33 00:01:34,109 --> 00:01:36,939 Then this orange line, that's perpendicular to the yellow 34 00:01:36,939 --> 00:01:42,000 line, is going to have a slope of negative 1 over m. 35 00:01:42,000 --> 00:01:44,310 Their slopes are going to be the negative 36 00:01:44,310 --> 00:01:46,689 inverse of each other. 37 00:01:46,689 --> 00:01:50,409 Now, given this information, let's look at a bunch of lines 38 00:01:50,409 --> 00:01:52,599 and figure out if they're parallel, if they're 39 00:01:52,599 --> 00:01:54,979 perpendicular, or if they are neither. 40 00:01:54,980 --> 00:01:58,570 And to do that, we just have to keep looking at the slopes. 41 00:01:58,569 --> 00:02:00,689 So let's see, they say one line passes through the 42 00:02:00,689 --> 00:02:05,310 points, 4, negative 3, and negative 8, 0. 43 00:02:05,310 --> 00:02:10,039 Another line passes through the points, negative 1, 44 00:02:10,038 --> 00:02:12,669 negative 1, and negative 2, 6. 45 00:02:12,669 --> 00:02:14,719 So let's figure out the slopes of each of these lines. 46 00:02:14,719 --> 00:02:16,680 So I'll first do this one in pink. 47 00:02:16,680 --> 00:02:19,700 So this slope right here, so line 1, so I'll 48 00:02:19,699 --> 00:02:21,789 call it slope 1. 49 00:02:21,789 --> 00:02:25,780 Slope 1 is, let's just say it is-- well, I'll take this as 50 00:02:25,780 --> 00:02:26,590 the finishing point. 51 00:02:26,590 --> 00:02:31,000 So negative 3 minus 0-- remember change in y-- 52 00:02:31,000 --> 00:02:41,129 negative 3 minus 0, over 4 minus negative 8. 53 00:02:41,129 --> 00:02:45,250 So this is equal to negative 3 over-- this is the same thing 54 00:02:45,250 --> 00:02:49,180 as 4 plus 8-- negative 3 over 12, which is 55 00:02:49,180 --> 00:02:52,219 equal to negative 1/4. 56 00:02:52,219 --> 00:02:55,750 Divide the numerator and the denominator by 3. 57 00:02:55,750 --> 00:02:56,610 That's this line. 58 00:02:56,610 --> 00:02:57,760 That's the first line. 59 00:02:57,759 --> 00:03:00,870 Now, what about the second line? 60 00:03:00,870 --> 00:03:05,180 The slope for that second line is, well, let's take, here, 61 00:03:05,180 --> 00:03:16,780 negative 1 minus 6, over negative 1 minus negative 2 is 62 00:03:16,780 --> 00:03:23,229 equal to-- negative 1 minus 6 is negative 7, over negative 1 63 00:03:23,229 --> 00:03:24,619 minus negative 2. 64 00:03:24,620 --> 00:03:26,750 That's the same thing as negative 1 plus 2. 65 00:03:26,750 --> 00:03:28,250 Well, that's just 1. 66 00:03:28,250 --> 00:03:32,300 So the slope here is negative 7. 67 00:03:32,300 --> 00:03:35,410 So here, their slopes are neither equal-- so they're not 68 00:03:35,409 --> 00:03:37,389 parallel-- nor are they the negative 69 00:03:37,389 --> 00:03:38,439 inverse of each other. 70 00:03:38,439 --> 00:03:39,180 So this is neither. 71 00:03:39,180 --> 00:03:42,020 This is neither parallel nor perpendicular. 72 00:03:42,020 --> 00:03:45,350 73 00:03:45,349 --> 00:03:48,349 So these two lines, they intersect, but they're not 74 00:03:48,349 --> 00:03:52,739 going to intersect at a 90 degree angle. 75 00:03:52,740 --> 00:03:55,210 Let's do a couple more of these. 76 00:03:55,210 --> 00:03:59,540 So I have here, once again, one line passing through these 77 00:03:59,539 --> 00:04:04,329 points, and then another line passing through these points. 78 00:04:04,330 --> 00:04:06,770 So let's just look at their slopes. 79 00:04:06,770 --> 00:04:11,010 So this one in the green, what's the slope? 80 00:04:11,009 --> 00:04:14,012 The slope of the green one, I'll call that the first line. 81 00:04:14,012 --> 00:04:16,329 We could say, let's see, change in y. 82 00:04:16,329 --> 00:04:23,030 So we could do negative 2 minus 14, over-- I did 83 00:04:23,029 --> 00:04:26,119 negative 2 first, so I'll do 1 first-- over 1 84 00:04:26,120 --> 00:04:29,090 minus negative 3. 85 00:04:29,089 --> 00:04:33,139 So negative 2 minus 14 is negative 16. 86 00:04:33,139 --> 00:04:36,199 1 minus negative 3 is the same things as 1 plus 3. 87 00:04:36,199 --> 00:04:37,459 That's over 4. 88 00:04:37,459 --> 00:04:40,000 So this is negative 4. 89 00:04:40,000 --> 00:04:43,610 Now, what's the slope of that second line right there? 90 00:04:43,610 --> 00:04:47,420 So we have the slope of that second line. 91 00:04:47,420 --> 00:04:57,069 Let's say 5 minus negative 3, that's our change in y, over 92 00:04:57,069 --> 00:05:01,490 negative 2 minus 0. 93 00:05:01,490 --> 00:05:04,329 So this is equal to 5 minus negative 3. 94 00:05:04,329 --> 00:05:06,539 That's the same thing as 5 plus 3. 95 00:05:06,540 --> 00:05:08,210 That's 8. 96 00:05:08,209 --> 00:05:11,459 And then negative 2 minus 0 is negative 2. 97 00:05:11,459 --> 00:05:14,699 So this is also equal to negative 4. 98 00:05:14,699 --> 00:05:16,279 So these two lines are parallel. 99 00:05:16,279 --> 00:05:20,319 100 00:05:20,319 --> 00:05:23,659 They have the exact same slope. 101 00:05:23,660 --> 00:05:25,890 And I encourage you to find the equations of both of these 102 00:05:25,889 --> 00:05:28,789 lines and graph both of these lines, and verify for yourself 103 00:05:28,790 --> 00:05:32,060 that they are indeed parallel. 104 00:05:32,060 --> 00:05:34,750 Let's do this one. 105 00:05:34,750 --> 00:05:37,209 Once again, this is just an exercise in finding slope. 106 00:05:37,209 --> 00:05:39,189 So this first line has those points. 107 00:05:39,189 --> 00:05:41,079 Let's figure out its slope. 108 00:05:41,079 --> 00:05:43,479 The slope of this first line, one line passes 109 00:05:43,480 --> 00:05:44,720 through these points. 110 00:05:44,720 --> 00:05:51,480 So let's see, 3 minus negative 3, that's our change in y, 111 00:05:51,480 --> 00:05:56,240 over 3 minus negative 6. 112 00:05:56,240 --> 00:06:00,000 So this is the same thing as 3 plus 3, which is 6, over 3 113 00:06:00,000 --> 00:06:02,120 plus 6, which is 9. 114 00:06:02,120 --> 00:06:05,300 So this first line has a slope of 2/3. 115 00:06:05,300 --> 00:06:07,590 What is the second line's slope? 116 00:06:07,589 --> 00:06:10,229 This was the second line there, that's the other line 117 00:06:10,230 --> 00:06:11,890 passing through these points. 118 00:06:11,889 --> 00:06:15,550 So the other line's slope, let's see, we could say 119 00:06:15,550 --> 00:06:27,340 negative 8 minus 4, over 2 minus negative 6. 120 00:06:27,339 --> 00:06:28,719 So what is this equal to? 121 00:06:28,720 --> 00:06:32,150 Negative 8 minus 4 is negative 12. 122 00:06:32,149 --> 00:06:36,139 2 minus negative 6, that's the same thing as 2 plus 6. 123 00:06:36,139 --> 00:06:37,610 The negatives cancel out. 124 00:06:37,610 --> 00:06:41,259 So it's negative 12 over 8, which is the same thing if we 125 00:06:41,259 --> 00:06:44,069 divide the numerator and the denominator by 4, that's 126 00:06:44,069 --> 00:06:45,319 negative 3/2. 127 00:06:45,319 --> 00:06:47,930 128 00:06:47,930 --> 00:06:50,120 Notice, these guys are the negative 129 00:06:50,120 --> 00:06:51,740 inverse of each other. 130 00:06:51,740 --> 00:06:57,850 If I take negative 1 over 2/3, that is equal to negative 1 131 00:06:57,850 --> 00:07:02,870 times 3/2, which is equal to negative 3/2. 132 00:07:02,870 --> 00:07:05,350 These guys are the negative inverses of each other. 133 00:07:05,350 --> 00:07:07,300 You swap the numerator and the denominator, make them 134 00:07:07,300 --> 00:07:09,430 negative, and they become equal to each other. 135 00:07:09,430 --> 00:07:14,579 So these two lines are perpendicular. 136 00:07:14,579 --> 00:07:18,149 And I encourage you to find the equations-- I already got 137 00:07:18,149 --> 00:07:20,560 the slopes for you-- but find the equations of both of these 138 00:07:20,560 --> 00:07:24,089 lines, plot them, and verify for yourself that they are 139 00:07:24,089 --> 00:07:25,619 perpendicular. 140 00:07:25,620 --> 00:07:27,579 Let's do one more. 141 00:07:27,579 --> 00:07:32,889 Find the equation of a line perpendicular to this line 142 00:07:32,889 --> 00:07:35,589 that passes through the point 2 comma 8. 143 00:07:35,589 --> 00:07:37,439 So this first piece of information, that it's 144 00:07:37,439 --> 00:07:40,649 perpendicular to that line right over there, what does 145 00:07:40,649 --> 00:07:41,810 that tell us? 146 00:07:41,810 --> 00:07:44,329 Well, if it's perpendicular to this line, its slope has to be 147 00:07:44,329 --> 00:07:46,949 the negative inverse of 2/5. 148 00:07:46,949 --> 00:07:49,920 So its slope, the negative inverse of 2/5, the inverse of 149 00:07:49,920 --> 00:07:54,509 2/5 is-- let me do it in a better color, a nicer green. 150 00:07:54,509 --> 00:07:57,870 If this line's slope is negative 2/5, the equation of 151 00:07:57,870 --> 00:08:00,610 the line we have to figure out that's perpendicular, its 152 00:08:00,610 --> 00:08:01,970 slope is going to be the inverse. 153 00:08:01,970 --> 00:08:05,180 So instead of 2/5, it's going to be 5/2. 154 00:08:05,180 --> 00:08:07,579 Instead of being a negative, it's going to be a positive. 155 00:08:07,579 --> 00:08:11,169 So this is a negative inverse of negative 2/5. 156 00:08:11,170 --> 00:08:11,580 Right? 157 00:08:11,579 --> 00:08:13,240 You take the negative sign and it becomes positive. 158 00:08:13,240 --> 00:08:15,689 You swap the 5 and the 2, you get 5/2. 159 00:08:15,689 --> 00:08:19,600 So that is going to have to be our slope. 160 00:08:19,600 --> 00:08:22,430 And we can actually use the point-slope form right here. 161 00:08:22,430 --> 00:08:24,850 It goes through this point right there. 162 00:08:24,850 --> 00:08:26,580 So let's use point-slope form. 163 00:08:26,579 --> 00:08:32,220 y minus this y-value, which has to be on the line, is 164 00:08:32,220 --> 00:08:38,879 equal to our slope, 5/2 times x minus this x-value, the 165 00:08:38,879 --> 00:08:41,220 x-value when y is equal to 8. 166 00:08:41,220 --> 00:08:43,740 And this is the equation of the line in point-slope form. 167 00:08:43,740 --> 00:08:46,940 If you want to put it in slope-intercept form, you can 168 00:08:46,940 --> 00:08:49,960 just do a little bit of algebra, algebraic 169 00:08:49,960 --> 00:08:52,910 manipulation. y minus 8 is equal to-- let's distribute 170 00:08:52,909 --> 00:08:59,959 the 5/2-- so 5/2 x minus-- 5/2 times 2 is just-- 5. 171 00:08:59,960 --> 00:09:01,570 And then add 8 to both sides. 172 00:09:01,570 --> 00:09:05,900 You get y is equal to 5/2x. 173 00:09:05,899 --> 00:09:09,634 Add 8 to negative 5, so plus 3. 174 00:09:09,634 --> 00:09:12,340 And we are done.